Problem: Factor the following expression: $-3$ $x^2+$ $13$ $x$ $-12$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(-12)} &=& 36 \\ {a} + {b} &=& & & {13} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $36$ and add them together. The factors that add up to ${13}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${9}$ $ \begin{eqnarray} {ab} &=& ({4})({9}) &=& 36 \\ {a} + {b} &=& {4} + {9} &=& 13 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 +{4}x +{9}x {-12} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 +{4}x) + ({9}x {-12}) $ Factor out the common factors: $ x(-3x + 4) - 3(-3x + 4) $ Notice how $(-3x + 4)$ has become a common factor. Factor this out to find the answer. $(-3x + 4)(x - 3)$